The Pólya-Szegö principle and the anisotropic convex Lorentz-Sobolev inequality.

Shuai Liu,Binwu He

doi:10.1155/2014/875245

Abstract

An anisotropic convex Lorentz-Sobolev inequality is established, which extends Ludwig, Xiao, and Zhang's result to any norm from Euclidean norm, and the geometric analogue of this inequality is given. In addition, it implies that the (anisotropic) Pólya-Szegö principle is shown.

Highlights

The classical Polya-Szegoprinciple states that for p ≥ 1 the inequality ∫ Rn 󵄨󵄨󵄨󵄨∇f󵄨󵄨󵄨󵄨pdx ≥󵄨󵄨󵄨󵄨∇f⋆󵄨󵄨󵄨󵄨pdx (1)holds for every of functions onRfn∈thCat0∞a(rRe snm),owohtheraenCd0∞ha(vRenc)odmenpoactetssuthpeposertt and | ⋅ | is the standard Euclidean norm
FK denotes the convex symmetrization of f, that is, a function whose level sets have the same measure as the level sets of f and are dilates of the set K
A convex body is a compact convex set in Rn which is throughout assumed to contain the origin in its interior

Summary

Introduction

The classical Polya-Szegoprinciple (see, e.g., [1, 2]) states that for p ≥ 1 the inequality. Instead of approach using the classical technique on level sets is using the Lp convexification of level [f]t, sets their ⟨f⟩t This technique plays a fundamental role in the newly emerged affine Polya-Szegoprinciple (see, e.g., [4,5,6,7,8,9]). Where V denotes the Lebesgue measure on Rn with κn = V(B) = πn/2/Γ(1 + n/2) This inequality has a geometric analogue, namely, the following Lp isoperimetric inequality: for 1 < p < n, Sp (L) ≥ nκnp/nV(L)(n−p)/n,. It is shown that our inequality (5) implies the anisotropic Polya-Szegoprinciple (2) for 1 ≤ p ≠ n in Theorem 5 It is true in Euclidean case; that is, (3) implies (1) for 1 ≤ p ≠ n.

Background

The Anisotropic Convex Lorentz-Sobolev Inequality

The Pólya-Szegö Principle